In a list of $18$ numbers, four of the numbers are increased by $4$, and four of the numbers are increased by $5$. By how much is the mean increased?
Solution: The sum of all of the numbers is increased by $4\cdot 4 + 4\cdot 5 = 36$. Because the mean is the sum divided by $18$, the mean is increased by $\frac{36}{18} =\boxed{2}$.